It's a good question. My best guess at an answer is that they probably can't. Here's why:
As we all know, we have material in the writings of St. Paul that point to different gender arrangements. Egalitarians like to point to Galatians 3.28, "there is neither male nor female." Patriarchalists like to point to texts like 1 Timothy 2.12, "I do not permit a woman to teach or to have authority over a man."
Which of these visions should regulate the life of the church?
A simple way to cut to the chase is to frame the question eschatologically. That is, when the Kingdom comes in its fullness will there be egalitarian power relations between men and women as envisioned in Gal. 3.28, with all in submission to Jesus and to each other? Or will men still have authority over women in heaven as envisioned in 1 Tim. 2.12?
I expect that, in my church at least, a lot of the men (and women) who preach 1 Tim. 2.12 as normative for the church actually believe that Gal. 3.28 describes what heaven will look like. That is, I don't think most patriarchalists, at my church at least, think that men will have authority over women in heaven. They believe that men only have authority over women now, on the earth.
And if that's the case I return to my question: Can patriarchalists pray the Lord's Prayer?
Can they pray "Thy Kingdom come. Thy will be done, on earth as it is in heaven?" and mean it?